The Cohen-Macaulay Property of Affine Semigroup Rings in Dimension 2

Abstract

Let $k$ be a field and $x,y$ indeterminates over $k$. Let $R=k[x^a,x^{p_1}{y}^{s_1},\dots ,x^{p_t}{y}^{s_t},y^b]\subseteq k[x,y]$. We calculate the Hilbert polynomial of $(x^a,y^b)$. The multiplicity of this ideal provides part of a criterion for the ring to be Cohen-Macaulay. Next, we prove a simple numerical criterion for $R$ to be Cohen-Macaulay in the case when $t=2$. We also provide a simple algorithm which identifies the monomial $k$-basis of $R/(x^a,y^b)$. Finally, these simple results are specialized to the case of projective monomial curves in ${\mathbb{P}}^3$.